IEE5328 Nanodevice Transport Theory and Computational Tools Prof. Ming-Jer Chen Dept. Electronics Engineering National Chiao-Tung University May 1, 2013 Lecture 7: Effective Mobility in 2DEG and 2DHG of Long-Channel Thick Gate-Oxide Planar Bulk MOSFETs: Microscopic Calculation (Advanced Device Physics with emphasis on hands-on calculations) 1IEE5328 Prof. MJ Chen NCTU

Two-Dimensional Hole Gas IEE5328 Prof. MJ Chen NCTU2

3 Thick Oxides No Stress Stress

4 IEE5328 Prof. MJ Chen NCTU

5 Kubo-Greenwood Formula v x μ is the group velocity of subband μ along x-direction and f 0 is the equilibrium Fermi distribution. The mobility formula in electron case is no longer valid for the hole case due to the failure of the effective mass approximation. Thus, the hole mobility formula as derived from the Boltzmann transport equation (BTE) must be used:

6 Kubo-Greenwood Formula 1 st Subband Si 300K F S =1MV/cm

7 Physical Models ‚Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering ; ; ω q Longitudinal Transverse Acoustic Optical Si Phonon Dispersion H’ Hole-AC Phonon H’ Hole-OP Phonon ΔE≈1meV within 1/2 Brillouin zone ΔE≈61.2meV Phonon wave vector

8 Physical Models ‚Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering The acoustic deformation potential D ac, is strongly connected to Bir-Pikus deformation potentials. According to Lawaetz, D ac can be formulated as ; ; c 11, c 12, and c 44 are the elastic coefficients H’ Hole-AC Phonon Small vibration term Elastic scattering Isotropic approximation

9 Physical Models ‚Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering According to Wiley and Costato and Reggiani, the optical deformation potential D op can have the following formalism: ; ; Average sound velocity H’ Hole-OP Phonon Small vibration term Inelastic scattering (61.2meV) Isotropic approximation ω op : optical phonon frequency; n op : Bose occupation factor of optical phonons

10 Physical Models ‚Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering ; ; H’ Hole-SR Small vibration term Elastic scattering p+p+ p+p+ Gate Inversion Layer n-type Substrate H’ Hole-SR Anisotropic scattering

11 Physical Models ‚Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering ; ; Anisotropic surface roughness scattering rate

12 Physical Models ‚Acoustic Phonon, Optical Phonon, and Surface Roughness Scattering ; ; Isotropic SR scatteringAnisotropic SR scattering *A. T. Pham, C. Jungemann, and B. Meinerzhagen, “Microscopic modeling of hole inversion layer mobility in unstrained and uniaxially stressed Si on arbitrarily oriented substrates,” Solid-State Electronics, vol. 52, pp , May 2008.

13 ; ;

Two-Dimensional Electron Gas IEE5328 Prof. MJ Chen NCTU14

15 IEE5328 Prof. MJ Chen NCTU Under the momentum relaxation time approximation, Thick Oxides

IEE5328 Prof. MJ Chen NCTU16 Thick Oxides Gaussian model: Exponential model:

IEE5328 Prof. MJ Chen NCTU17 Thin Oxides

IEE5328 Prof. MJ Chen NCTU18 Thin Oxides

IEE5328 Prof. MJ Chen NCTU19

20 Inversion layer mobility in thick oxide MOSFETs can be limited to three primary scattering mechanisms:

 Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region - The scattering rate of ionized impurity in 3-D case can be presented by: - However, it is not the 2-D electron gas inside the MOSFET. Nano Electronics Physics NCTU 21 Ionized Impurity Scattering Model : the ionized impurity concentration : the Debye length can be written as, where n 0 is the 3-D density of the mobile carrier

 Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region at 2-D case - The momentum conservation in the z-direction of the 3-D case at the scattering process of 2-D carriers should be replaced by the integral as: Therefore, the scattering rate of ionized impurity scattering in 2-D case from m th subband to n th subband can be expressed as: Nano Electronics Physics NCTU 22 where H 2D and H 3D are the matrix elements for 2-D and 3-D scattering, respectively. :The average inversion layer thickness :the density of states for two dimensions :the density of states for three dimensions Ionized Impurity Scattering Model

- For intravalley phonon scattering model, the momentum- relaxation rate in the subband m th to n th : The total scattering rate in m th subband is determined by summing up within all the subbands: Nano Electronics Physics NCTU 23 Phonon Scattering Model D ac : deformation potential due to acoustic phonons S l : sound velocity ρ : crystal density W m,n : the form factor determined by the wave-functions of the m th subband and n th subbands

- For the intervalley phonon scattering model: (Incorrect Version) (1). From m th subband in twofold valleys to the n th subband in fourfold valleys: (2). From m th subband in fourfold valleys to the n th subband in twofold valleys: (3). From m th subband in fourfold valleys to the n th subband in fourfold valleys: (4). From m th subband in twofold valleys to the n th subband in twofold valleys : Nano Electronics Physics NCTU 24 Phonon Scattering Model where E k and D k are deformation energy and potential at k th intervalley phonon, and N k is the occupation number of k th intervalley phonon.

- The scattering rate for a Gaussian function is described as: Nano Electronics Physics NCTU 25 Surface Roughness Scattering Model Δ : rms height of the amplitude of surface roughness : correlation length of surface roughness

 Derivation of Two-Dimensional Mobility in the Universal Mobility Region - The scattering rates of the twofold and fourfold valley: The electron mobility by using the average energy within the 2DEG in the relaxation time approximation can be given as The total universal mobility averaged over the subband occupation is described by Nano Electronics Physics NCTU 26 Electron Mobility Model

 The physical parameters for phonon and surface-roughness electron mobility used for Si in this work Nano Electronics Physics NCTU 27 Electron Mobility Model

 The Universal mobility Curve - Universal electron mobility includes phonon scattering and surface roughness scattering, which is independent of process parameters, especially when plotted versus of high effective field (E eff ). Nano Electronics Physics NCTU 28 Electron Mobility Model : total inversion layer charge density :the surface concentration of the depletion charge

Electron Effective Mobility: Thick Oxides IEE5328 Prof. MJ Chen NCTU29 Current NEP-electron-mobility simulator was developed under the parabolic band approximation, the isotropic scattering approximation, the elastic scattering approximation (surface roughness and impurity scattering), and the momentum relaxation approximation.

30 Ionized Impurity Scattering The perturbing potential is the screened Coulomb potential: r: the distance from the scattering center L D: Debye length n: 3-D density of the mobile carriers and equal to N inv /Z av Through the Fermi’s Golden Rule, the relaxation time due to ionized impurity scattering is: N I : 3-D impurity concentration, which is about equal to N sub Here, q means the elementary charge.

31 Impurity Scattering The momentum relaxation time for scattering of 2-D carriers from u th subband to v th subband: P.S. Only consider intra-subband K. Hirakawa and H. Sakaki, “Mobility of the two-dimensional electron gas at selectively doped n-type Al x Ga 1-x As/GaAs heterojunctions with controlled electron concentrations,” Phys. Rev. B, Condens. Matter, vol. 33, no. 12, , Jun M. Lundstrom, “Fundamentals of carrier transport,” Cambridge University Press, 2000.

32 Phonon Scattering 1. intravalley phonon scattering model: (acoustic phonon) the momentum-relaxation time for scattering from the u th subband to the v th subband k B : Boltzmann constant, D ac : deformation potential due to acoustic phonons ρ : crystal density, S ι : sound velocity U(x): step function S. Takagi, J. L. Hoyt, J. J. Welser, and J. F. Gibbons, “Comparative study of phonon-limited mobility of two-dimensional electrons in strained and unstrained Si metal-oxide-semiconductor field-effect transistors,” J. Appl. Phys., vol. 80, no. 3, pp , Aug D.Esseni, A. Abramo, L. Selmi, and E. Sangiorgi, “Physically Based modeling of Low Field Electron Mobility in Ultrathin Single- and Double-Gate SOI n-MOSFETs”, IEEE Trans. Electron Devices, vol. 50, no.12, pp , K. Uchida, A. Kinoshita, and M. Saitoh, “Carrier transport in (110) nMOSFETs: Subband structures, non parabolicity, mobility characteristics, and uniaxial stress engineering,” in IEDM Tech. Dig., pp , 2006.

33 Phonon Scattering model (Correct Version) 2. intervalley phonon scattering model: (optical phonon) (a) From u th subband in twofold valleys to the v th subband in twofold valleys: (b) From u th subband in twofold valleys to the v th subband in fourfold valleys: (c)From u th subband in fourfold valleys to the v th subband in twofold valleys: (d)From u th subband in fourfold valleys to the v th subband in fourfold valleys:

34 22 44 44 44 44 f-type Acoustic phonon g-type

35 Surface Roughness Scattering P.S. Only consider intra-subband Yamakawa, H. Ueno, K. Taniguchi, C. Hamaguchi, K. Miyatsuji, K. Masaki, and U. Ravaioli, “Study of interface roughness dependence of electron mobility in Si inversion layers using the Monte Carlo method,” J. Appl. Phys., vol. 79, no. 2, pp , Jan D.Esseni, “On the Modeling of Surface Roughness Limited Mobility in SOI MOSFETs and its Correlation to the Transistor Effective Field”, IEEE TED, Vol.51 NO.3, pp , 2004.

36 Three-Dimensional Stress Effect on Electron Mobility